Blender 3D Rigging Bone Troubleshooting
First, select the arms and Clear the Parent.

Now select the arm again and Set Parent To Bone.

I think the vertex groups are not painted well.

I repainted the meshes but the result was not so good.

The best thing I could do is to reparent the objects.

See the image below: 



More:

It's a good practice, as soon as you unparented the objects, to delete all the vertex groups of that object.